Generalized Secrecy Outage Probability (GSOP)

• GSOP is a refined physical-layer security metric that quantifies the probability of achieving a specified secrecy level using fractional equivocation.
• It offers closed-form and asymptotic expressions applicable to relay networks, multi-antenna systems, and network-coded cooperative scenarios.
• GSOP guides system design by optimizing power allocation, antenna configuration, and cooperative protocols to balance information leakage and throughput.

The generalized secrecy outage probability (GSOP) is a refined metric in physical layer security that quantifies the probability that the level of secrecy achieved for a given transmission falls below a specified threshold, rather than simply indicating a binary success or failure based on classical secrecy outage criteria. GSOP explicitly accounts for scenarios where partial secrecy is acceptable, and quantifies the degree of information leakage in terms of fractional equivocation or related measures. This generalization provides a nuanced framework for the analysis and design of secure wireless systems, especially under realistic fading, diversity, and network environments.

1. Foundations and Definitions

The classic secrecy outage probability (SOP) is defined as the probability that the instantaneous secrecy capacity falls below a target secrecy rate. Mathematically, for instantaneous capacities $C_b$ at the legitimate receiver and $C_e$ at the eavesdropper, and target $R_s$:

$\text{SOP} = \Pr\{ [C_b - C_e]^+ < R_s \}$

GSOP extends this by evaluating the likelihood that the fractional equivocation (the fraction of message uncertainty remaining at the eavesdropper) is less than a desired threshold $\theta \in (0, 1]$:

$\text{GSOP} = \Pr\{ \Lambda < \theta \}$

where $\Lambda = H(M|Z^n)/H(M)$ denotes the fractional equivocation, $H(M)$ is the entropy of the message, and $H(M|Z^n)$ is the eavesdropper's remaining uncertainty after observing $n$ channel uses (He et al., 2016, Mora et al., 15 Sep 2025).

By varying $\theta$, GSOP interpolates between classical "perfect secrecy" ($\theta = 1$) and partial secrecy, quantifying how much information the eavesdropper can potentially glean.

2. Closed-Form Expressions and Asymptotic Behavior

Multiple works provide exact and approximate expressions for GSOP in diverse scenarios:

• Relay Networks: For selective decode-and-forward relaying, the closed-form GSOP is given for $N$ independently distributed relays under Rayleigh fading as

$$P_{\text{out}}(R) = \sum_{n=0}^N C(N, n) \left(-\frac{\lambda_m}{e^R \lambda_e + \lambda_m}\right)^n \exp\left(-\frac{n(e^R-1)}{\lambda_m}\right)$$

where $\lambda_m, \lambda_e$ are average SNRs of main and eavesdropper links (Sun et al., 2011).

• Diversity-Enhanced Multi-Antenna Systems: Under generalized fading (e.g., multicluster fluctuating two-ray), GSOP is linked to the cumulative distribution function (CDF) of an auxiliary variable $\Phi$ dependent on the main and wiretap SNRs (Mora et al., 15 Sep 2025):

$\text{GSOP} = F_\Phi(2^{\theta R_s})$

with further closed-form approximations and asymptotics:

$$\text{GSOP}^{(\infty)} = \mathcal{G}_c\, (\bar{\gamma}_B)^{-\mathcal{G}_d}$$

where $\mathcal{G}_d$ reflects the diversity gain, determined by the product of MRC branches and number of clusters.

• Network Coded Cooperation: In multi-source cooperative networks with network coding, the GSOP (for partial CSI) is given (see (Rebelatto et al., 2014), eqn. (21)) as

$$S_{\text{GNC}}^{\text{CSI}} \approx (M k_2 + 1) \sum_{i=0}^{M+k_2} (-1)^i {M+k_2 \choose i} \exp\left(-\frac{\xi-1}{\bar{\gamma}_D i}\right) B\left(\cdots\right)$$

where code rate, diversity, and coding gain are explicit in the expression.

The asymptotic (high-SNR) regime is particularly important. For example, in the relay selection case, as $\lambda_m,\lambda_e \to \infty$ with fixed ratio $\kappa = \lambda_m/\lambda_e$: $$P_{\mathrm{out}}^a(R) = \left(\frac{e^R}{e^R + \kappa}\right)^N$$ (Sun et al., 2011) demonstrating that increasing relays or SNR improves secrecy performance polynomially according to the diversity order.

3. Partial Secrecy, Fractional Equivocation, and Information Leakage

GSOP enables a graded view of secrecy by adopting the metric of fractional equivocation $\Lambda$ (He et al., 2016). Classical SOP corresponds to $\Pr\{\Lambda < 1\}$, while GSOP for arbitrary $\theta$ allows systems to specify tolerable levels of information leakage. Two further metrics are:

• Average Fractional Equivocation: $\bar{\Lambda} = \mathbb{E}\{\Lambda\}$ acts as a lower bound on the eavesdropper's decoding error probability.
• Average Information Leakage Rate: $\mathcal{R}_L = \mathbb{E}[(1-\Lambda) R]$, quantifying the average rate at which the eavesdropper gains information.

These measures are practical for system design in scenarios with finite blocklength or time-varying environments where perfect secrecy may be either too strict or unattainable.

4. Influence of Fading, Diversity, and Network Architecture

The impact of channel statistics and architectural choices on GSOP is multifaceted:

• Diversity: The diversity order (i.e., the exponent $\mathcal{G}_d$ in the high-SNR decay of GSOP) is tightly coupled to the number of independent fading paths, antenna elements (MRC), or network dimensions. For instance, GSOP decays as $(1/\bar{\gamma}_B)^{\mu_B L_B}$ in MFTR fading for $L_B$ MRC at the legitimate receiver (Mora et al., 15 Sep 2025).
• Eavesdropper Diversity: Adding antennas at the eavesdropper improves the wiretap channel and degrades secrecy, but the secrecy diversity order is determined by the legitimate link's diversity parameters.
• Network Structure: Cooperative jamming, network coding, or opportunistic relay/pair selection strategies provide multiplicative scaling improvements in GSOP by introducing further diversity. In multiuser or multicasting settings, the minimum among legitimate links and the maximum among eavesdropper links jointly dictate the GSOP (Hossen et al., 2021).
• Channel Models: Results hold for generalized fading laws, e.g., fluctuating two-ray, $\alpha$-$\mu$, composite shadowed, and Weibull, allowing general conclusions across wireless deployments.

5. Resource Allocation and System Design Implications

The explicit characterization of GSOP supports several design and optimization tasks:

• Power Allocation: For systems permitting partial secrecy, target GSOP levels can be directly mapped to minimum required SNRs or optimized power budgets (Luo et al., 2012, Thapar et al., 2023).
• Antenna Configuration: The analysis can predict diminishing returns for added diversity or quantify the minimum required number of transmit or receive antennas for a prescribed GSOP.
• Network Coding Parameter Tuning: Code rate and diversity order can be adjusted to control the GSOP floor and decay rate with SNR (Rebelatto et al., 2014).
• Threshold Selection: Choice of fractional equivocation threshold $\theta$ provides a tunable security-vs-throughput tradeoff, facilitating adaptation to different application requirements or threat models (Mora et al., 15 Sep 2025, He et al., 2016).
• Scheduling and Cooperation: Cooperative and opportunistic protocols (e.g., best-node selection in wireless sensor networks) achieve lower GSOP with increasing network size, unless limited by outdated CSI or link quality (Jameel et al., 2019).

6. Practical Considerations and Validation

Closed-form and asymptotic GSOP expressions are amenable to efficient numerical evaluation even in high-diversity regimes, as their complexity can be made independent of diversity order by construction (Mora et al., 15 Sep 2025). Monte Carlo simulations across multiple works confirm the tightness of both exact and asymptotic approaches, validating their use for rapid network planning and real-time assessment. The independence of expression complexity with respect to receive diversity (provided by, e.g., MRC) facilitates their inclusion in hardware-constrained devices and emerging dense wireless networks.

Practical guidelines derived include:

• Leveraging MRC or cooperative strategies at the legitimate receiver substantially increases secrecy diversity order, efficiently tightening GSOP.
• Sophisticated network coding and scheduling further suppress GSOP, especially in the absence of instantaneous CSI at the transmitter.
• The use of partial secrecy metrics is particularly relevant for IoT and next-generation networks where computation and energy constraints preclude traditional cryptographic approaches.

7. Summary Table: Representative GSOP Formulas and Their Features

Scenario / Model	GSOP / SOP Formula (Representative)	Key Parameters / Diversity Order
Selective DF Relaying (Rayleigh)	$$P_{\text{out}}(R) = \sum_{n=0}^N C(N, n) [ -\lambda_m / (e^R \lambda_e + \lambda_m) ]^n e^{-n(e^R - 1)/\lambda_m}$$	$N$ relays; $\lambda_m, \lambda_e$
MFTR Fading, Partial Secrecy (GSOP)	$\text{GSOP} = F_\Phi(2^{\theta R_s})$ (see closed-form in (Mora et al., 15 Sep 2025))	Diversity: $\mu_B L_B$
Network Coding Coop. (Partial CSI)	$$S_{\text{GNC}}^{\text{CSI}} \approx (M k_2 + 1) \sum_{i=0}^{M+k_2} (\cdots)$$	$M+k_2$; code rate $R_\text{GNC}$
Asymptotics (e.g. GSOP decay rate)	$$\text{GSOP}^{(\infty)} \sim (\mathcal{G}_c/\bar{\gamma}_B)^{\mathcal{G}_d}$$	As above

The diversity gain $\mathcal{G}_d$ summarizes how rapidly GSOP decreases with SNR, scaling with system and propagation parameters.

8. Significance and Research Directions

GSOP unifies and generalizes several physical-layer security metrics, enabling a continuum between strict secrecy guarantees and tolerable information leakage. It is instrumental in performance benchmarks for emerging systems under heterogeneous fading and network conditions. Ongoing research explores dynamic optimization of GSOP under stringent energy or latency constraints, distributed resource allocation in dense cooperative settings (Zamanipour, 2022), and reinforcement learning-based secrecy policy adaptation in adversarial or non-stationary environments.

In conclusion, the GSOP framework provides both a rigorous analytical tool for evaluating the efficacy of physical-layer security techniques and a pragmatic foundation for adaptive, resource-aware secure system design under real-world fading. It bridges the gap between information-theoretic abstractions and performance-centric requirements in modern wireless networks, particularly as partial secrecy becomes a central paradigm for practical security provisioning.